Finite Group Databasebeta

← #10845277

Subgroups — #10845277

Order 768.

824 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2367C211yesmaximal, characteristic
2C2×#5054C211yesmaximal, characteristic
3C2×#5055C211yesmaximal, characteristic
4#73722C211yesmaximal, characteristic, complemented
5#71632C211yesmaximal, characteristic, complemented
6#71627C211yesmaximal, characteristic, complemented
7#71612C211yesmaximal, characteristic
8#71605C211yesmaximal, characteristic
9#69681C211yesmaximal, characteristic, complemented
10#69671C211yesmaximal, characteristic, complemented
11#69651C211yesmaximal, characteristic
12#69641C211yesmaximal, characteristic
13#69489C211yesmaximal, characteristic, complemented
14#69485C211yesmaximal, characteristic, complemented
15#69475C211yesmaximal, characteristic, complemented
16#25337—31nomaximal, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C22×#214C2211yescharacteristic
22C2×#869C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C3×#391C2211yescharacteristic
25C3×#390C2211yescharacteristic
26C3×#388—21no—
27C3×#388—21no—
28C3×#385—21no—
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×C4×C4⋊C4C2211yescharacteristic
31C3×C4×C22⋊C4—21no—
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38#5055—24no—
39#5054—24no—
40#4973—21no—
41#4973—21no—
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44#4936C2211yescharacteristic
45#4936C2211yescharacteristic
46#4929C2211yescharacteristic
47#4929C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51C2×#803C2211yescharacteristic
52#4659—21no—
53#4659—21no—
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#489—31no—
63C2×#464—31no—
64#2568—31no—
65#2367D311yescharacteristic
66#2074—31no—
67#2073—31no—
68#2056—31no—
69#2055—31no—
70#1987—31no—
71#1940—31no—
72#1878—31no—
73#1877—31no—
74#1805—31no—
75#1804—31no—
76#1763—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C22×Dic6—21no—
81C22×Dic6C2311yescharacteristic
82C3×#124—22no—
83C3×#124D414yes—
84C3×C4×Q8—22no—
85C3×C4×Q8—41no—
86C3×C4×D4—22no—
87C3×C4×D4—41no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C42—21no—
99C2×C3×C42—21no—
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×#216—22no—
104C2×#216—22no—
105C2×#216—41no—
106C2×#214—22no—
107C2×#214—22no—
108C2×#214—41no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3—21no—
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#869—22no—
120#869—22no—
121C4×Dic6—22no—
122C4×Dic6—22no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95D411yescharacteristic
127C3×#95D411yescharacteristic
128#835—22no—
129#835—22no—
130#835—22no—
131#832—22no—
132#832—22no—
133#832—22no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142#810—22no—
143#810—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)—41no—
146C4⋊(C3⋊C8)—41no—
147#803—22no—
148C22×Q16—31no—
149C22×SD16—31no—
150C2×C4⋊Q8—31no—
151C2×#124C2×D311yescharacteristic
152C2×C4×Q8—31no—
153C2×C4×Q8C2×D311yescharacteristic
154C2×C4×D4C2×D311yescharacteristic
155#489—64no—
156#464—64no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#97—31no—
169#400—61no—
170#400—61no—
171#391—31no—
172#391—31no—
173#391—61no—
174#391—61no—
175#391C2×D311yescharacteristic
176#390C2×D311yescharacteristic
177#388—21no—
178#388—21no—
179#385—21no—
180#382—31no—
181#382—31no—
182C4×C4⋊C4C2×D311yescharacteristic
183C4×C4⋊C4C2×D311yescharacteristic
184C4×C22⋊C4—21no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C24×C3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8C2×D412yes—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4C2×D412yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—42no—
210C22×C3×C4—42no—
211C22×C3×C4—81no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—42no—
229C2×Dic6—42no—
230C2×Dic6—42no—
231C2×Dic6#4112yes—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—41no—
236C3×C4⋊C4—41no—
237C3×C4⋊C4—42no—
238C3×C4⋊C4#4112yes—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—42no—
242C3×C22⋊C4—42no—
243C3×C22⋊C4—81no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—42no—
249C3×C42—81no—
250C3×C42C2×D412yes—
251C3×C42#4112yes—
252#216—44no—
253#214—44no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—42no—
261Dic3⋊C4—42no—
262C4×Dic3—42no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C22×Q8—31no—
276C22×Q8—61no—
277C22×Q8C22×D311yescharacteristic
278C22×D4C22×D311yescharacteristic
279C23×C4—21no—
280C2×Q16—62no—
281C2×Q16—62no—
282C2×Q16—121no—
283C2×SD16—62no—
284C2×SD16—62no—
285C2×SD16—121no—
286C22×C8—31no—
287C22×C8—31no—
288C22×C8—31no—
289C22×C8—31no—
290C4⋊Q8—62no—
291C4⋊Q8—62no—
292#124—22no—
293#124D1214yes—
294C4×Q8—22no—
295C4×Q8—41no—
296C4×Q8—62no—
297C4×Q8—62no—
298C4×D4—22no—
299C4×D4—41no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4#6711yescharacteristic
307C2×C4⋊C4#6711yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319C2×C42C22×D311yescharacteristic
320C2×C42C22×D311yescharacteristic
321C4⋊C8—62no—
322C4⋊C8—121no—
323C4⋊C8—121no—
324#103—62no—
325#103—62no—
326#103—62no—
327#103—62no—
328#103—62no—
329#102—62no—
330#102—62no—
331#102—62no—
332#97—62no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95#6711yescharacteristic
345#95#6711yescharacteristic
346C23×C3—22no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3C22×D411yescharacteristic
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×D4—22no—
359C3×D4—22no—
360C3×D4—41no—
361C3×D4—41no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—22no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—84no—
388C2×C3×C4#12112yes—
389C2×C3×C4#12112yes—
390C2×C3×C4C22×D411yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4H(5,𝔽2)11yescharacteristic
395C2×C3×C42−1+411yescharacteristic
396C2×Dic3—22no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408Dic6—22no—
409Dic6—22no—
410Dic6—41no—
411Dic6—41no—
412Dic6—84no—
413C3⋊C8—41no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C3⋊C8—41no—
417C24—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—32no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—122no—
430C2×Q8—122no—
431C2×Q8—122no—
432C2×Q8C2×D1212yes—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4C2×D1212yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—81no—
449C22×C4C2×#6711yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C2×#6711yescharacteristic
452C22×C4C2×#6711yescharacteristic
453C22×C4C2×#6711yescharacteristic
454C22×C4C2×#6711yescharacteristic
455C22×C4C23×D311yescharacteristic
456Q16—124no—
457SD16—124no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—42no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—122no—
483C4⋊C4—122no—
484C4⋊C4#23712yes—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—81no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—42no—
495C42—81no—
496C42—122no—
497C42C2×D1212yes—
498C42#23512yes—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—81no—
503C22×C3—81no—
504C22×C3—81no—
505C22×C3—81no—
506C22×C3C2×#12111yescharacteristic
507C22×C3#54611yescharacteristic
508C22×C3#54811yescharacteristic
509C22×C3#57511yescharacteristic
510C22×C3#57511yescharacteristic
511C22×C3#57711yescharacteristic
512C22×C3#57811yescharacteristic
513C3×C4—22no—
514C3×C4—22no—
515C3×C4—42no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—42no—
519C3×C4—41no—
520C3×C4—41no—
521C3×C4—81no—
522C3×C4#52911yescharacteristic
523C3×C4#54211yescharacteristic
524C3×C4#54611yescharacteristic
525C3×C4#54711yescharacteristic
526Dic3—42no—
527Dic3—81no—
528Dic3—81no—
529Dic3—81no—
530C23—22no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23C22×#6711yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—41no—
541Q8—41no—
542Q8—62no—
543Q8—62no—
544Q8—121no—
545Q8—121no—
546Q8—244no—
547D4—22no—
548D4—22no—
549D4—41no—
550D4—41no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—62no—
577C2×C4—84no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4#93012yes—
590C2×C4#93012yes—
591C2×C4C22×D1211yescharacteristic
592C2×C4C2×#23511yescharacteristic
593C2×C4C2×#23711yescharacteristic
594C2×C4#100411yescharacteristic
595C2×C4D3×#4111yescharacteristic
596C2×C4#101011yescharacteristic
597C8—121no—
598C8—121no—
599C8—121no—
600C8—121no—
601C2×C3—82no—
602C2×C3#242711yescharacteristic
603C2×C3#304011yescharacteristic
604C2×C3#304611yescharacteristic
605C2×C3#331111yescharacteristic
606C2×C3#331311yescharacteristic
607C2×C3#337111yescharacteristic
608C2×C3#337711yescharacteristic
609C22—42no—
610C22—42no—
611C22—42no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22C2×#93011yescharacteristic
617C22#580611yescharacteristic
618C22#582211yescharacteristic
619C22#582211yescharacteristic
620C22#583111yescharacteristic
621C22#583611yescharacteristic
622C22#583811yescharacteristic
623C4—22no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—42no—
628C4—41no—
629C4—41no—
630C4—42no—
631C4—81no—
632C4—122no—
633C4—241no—
634C4—241no—
635C4—241no—
636C4#555411yescharacteristic
637C4#555511yescharacteristic
638C4#555711yescharacteristic
639C4#556111yescharacteristic
640C3#2533711yescharacteristic, complemented, Sylow
641C2—82no—
642C2#8240811yescharacteristic
643C2#8252711yescharacteristic
644C2#8257211yescharacteristic
645C2#8349711yescharacteristic
646C2#8353711yescharacteristic
647C2#8367811yescharacteristic
648C2#8371811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2367C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2367D3
hypocenterC3#25337
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #73722C2semidirect
5. #71632C2semidirect
6. #71627C2semidirect
9. #69681C2semidirect
10. #69671C2semidirect
13. #69489C2semidirect
14. #69485C2semidirect
15. #69475C2semidirect
640. C3#25337semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#23672
C2×#50542
C2×#50552
#737222
#716322
#716272
#716122
#716052
#696812
#696712
#696512
#696412
#694892
#694852
#694752
#253373